Congruence and non-congruence level structures on elliptic curves: a hands-on tour of the modular tower

Classically, one puts an algebraic structure on certain "congruence" quotients of the upper half plane by interpreting them as spaces parametrising elliptic curves with certain level structures on their torsion subgroups. However, the non-congruence quotients don't admit such a straightforward description.

We will sketch the classical theory of congruence modular curves and level structures, and then discuss a preprint by W. Chen which extends the above notions to non-congruence modular curves by considering so-called Teichmueller level structures on the fundamental groups of punctured elliptic curves.